How to obtain a 2D-coordinate system from two family of curves?

it is known that if we have a curvilinear coordinate system in ℝ2 like [itex]x=x(u,v)[/itex], [itex]y=y(u,v)[/itex], and we keep one coordinate fixed, say [itex]v=\lambda [/itex], we obtain a family of one-dimensional curves [itex]C_{\lambda}(u)=\left( x(u,\lambda),y(u,\lambda) \right)[/itex]. The analogous argument holds for the other coordinate u. These family of curves are sometimes called coordinate lines, or level curves.

My question is: if I am given two family of curves [itex]C_v(u)[/itex] and [itex]C_u(v)[/itex] is it possible to obtain the system of curvilinear coordinates [itex]x(u,v)[/itex], [itex]y(u,v)[/itex] that generated them?

Yes, if we have a family of circles [itex]C_r(\theta)=\left( r\cos\theta, r\sin\theta \right)[/itex] for some [itex]r\in \mathbb{R}^+[/itex], and a family of straight lines passing through the origin [itex]C_\theta(r)=\left( r\cos\theta, r\sin\theta \right)[/itex] for some [itex]\theta\in[0,2\pi)[/itex] the solution is quite trivial.

I was interested more in a general procedure or simply a strategy that I could follow to solve this kind of problem.

If we cannot answer the general question then let's try at least a less trivial example I was unable to solve like this one: we have two families of "parallel" exponential curves, the first family is [itex]C_\lambda(u) = (u, \; e^u +\lambda)[/itex] for some fixed real scalars v, and the other family is [itex]C_k(v) = (e^{-v} + k, \; v)[/itex] for some real k.
I was unable to obtain two functions x(u,v) , y(u,v) such that [itex]C_\lambda(u) = (x(u,\lambda), \; y(u,\lambda))[/itex] and [itex]C_k(v)=(x(k,v),\; y(k,v))[/itex]